Hello Kerry, I would start with Mayo Library Local Studies and see if they have anything, http://www.mayolibrary.ie/en/LocalStudies/ you should mention the years involved etc otherwise the search period will bring too many hits. There is a subscription site British Newspaper Archive which is probably best pre 1921. I searched for Belmullet History online and got a number of hits and I notice there and on the library site there are a number of books written by Birkett ladies in more recent times.
I assume when he donated to Trinity it was to the Museum Building (beautiful inside by the way) there was a book published last year on the history of the Museum Building and is available here https://www.fourcourtspress.ie/books/2019/the-museum-building-of-trinity-college-dublin/ you could email the college to see if it is in it or the publishers.
A longer shot is the College of General Practitioners in Ireland, https://www.icgp.ie/ I doubt this will be fruitful though as not sure when it was incorporated. The Irish Times have an online archive from 1857, again subscription site, can join for a day or a week etc https://www.irishtimes.com/
There are also a number of Mayo Facebook pages on genealogy and history so you could search for some and join and ask the question.
Good Luck,
Pat